Pointwise error estimates and asymptotic error expansion inequalities for the finite element method on irregular grids. II: Interior estimates (Q2706362)

This paper is the second in a series of papers devoted to sharpening estimates of finite element solutions of second order partial differential equations. The first paper in this series [\textit{A. H. Schatz}, Math. Comput. 67, No. 223, 877-899 (1998; Zbl 0905.65105)] focused on \(L_{\infty}\) error estimates for solutions of the Neumann problem. In this paper, the author presents \(L_{\infty}(B_h(x))\) estimates, where \(B_h(x)\) is a ball of radius \(h\) about the interior point \(x\) and he does not confine himself to Neumann boundary conditions. As in the previous paper, the norms are weighted by the function \(\sigma^s_{x,h}(y)=({h\over|x-y|+h})^s\) so that they depend strongly on local behavior. NEWLINENEWLINENEWLINEThe underlying problem is defined on a domain \(\Omega\), and the author supposes a domain \(\Omega_d\), compactly contained in \(\Omega\) and with \(\text{ dist}(\partial\Omega_d,\Omega)=d\). The weak formulation is then restated to involve integration over \(\Omega_d\) rather than \(\Omega\). In this manner, consideration of boundary conditions on \(\partial\Omega\) can be avoided. NEWLINENEWLINENEWLINEAs in the previous paper, the author presents four related results. Let \(u\) denote the exact solution and \(u_h\) its approximation on \(\Omega_d\). The first result involves an estimate of \(\|u-u_h\|_{L_\infty(B_h(x))}\) in terms of: (1) the weighted \(W^1_\infty(\Omega_d)\) norm of the best finite element approximation to \(u\); (2) a term involving a negative power of \(d\) times \(\|u-u_h\|_{W_p^{-t}(\Omega_d)}\) for positive \(p\) and \(t\); and, (3) terms involving both weighted and unweighted norms of the forcing function, \(F\). The second result is similar, but involves weighted error estimates of the derivative, \(\|u-u_h\|_{W^1_\infty(B_h(x))}\). NEWLINENEWLINENEWLINEThe third and fourth results are similar to the first and second, but have the approximation error term replaced by an expansion of the error in powers of \(h\) multiplied by derivatives of \(u\) evaluated at one point near \(x\) plus a higher order remainder term. Taylor's theorem is used in the replacement process. These error expansion terms are the basis of promised results for Richardson extrapolation, superconvergence, and a posteriori estimates.